Higher order elliptic equations in weighted Banach spaces

TL;DR

This paper investigates higher order elliptic equations with non-smooth coefficients in weighted Banach spaces, establishing local solvability, a priori estimates, and Fredholm properties under certain boundedness assumptions.

Contribution

It introduces a framework for analyzing elliptic equations in weighted Banach function spaces, proving solvability and regularity results that extend classical theories.

Findings

Established local solvability in Sobolev-Banach function spaces.

Proved interior Schauder type a priori estimates for elliptic operators.

Demonstrated Fredholmness of the elliptic operator in weighted BFSs.

Abstract

We consider higher order linear, uniformly elliptic equations with non-smooth coefficients in Banach-Sobolev spaces generated by weighted general Banach Function Space (BFS). Supposing boundedness of the Hardy-Littlewood Maximal and Calderon-Zygmund singular operators in BFSs we obtain local solvability in the Sobolev-BFS and establish interior Schauder type a priori estimates for the. elliptic operator. These results will be used in order to obtain Fredholmness of the operator under consideration in weighted BFSs with suitable weight. In addition, we analyze some examples of weighted BFS that verify our assumptions and in which the corresponding Schauder type estimates and Fredholmness of the operator hold true.